Accepted Paper

Inserted: 22 sep 2017
Last Updated: 14 mar 2018

Journal: IMRN
Year: 2017

Abstract:

We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions.

The method is based on a geometric formula of Poincar\'e type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. {F}rom this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant.

As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano.

In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

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• Casten_Holland_Robin-REV.pdf